The use of two-point Taylor expansions in singular one-dimensional boundary value problems I

We consider the second-order linear differential equation (x+1)y″+f(x)y′+g(x)y=h(x) in the interval (−1,1) with initial conditions or boundary conditions (Dirichlet, Neumann or mixed Dirichlet-Neumann). The functions f(x), g(x) and h(x) are analytic in a Cassini disk Dr with foci at x=±1 containing the interval [−1,1]. Then, the end point of the interval x=−1 may be a regular singular point of the differential equation. The two-point Taylor expansion of the solution y(x) at the end points ±1 is used to study the space of analytic solutions in Dr of the differential equation, and to give a criterion for the existence and uniqueness of analytic solutions of the boundary value problem. This method is constructive and provides the two-point Taylor approximation of the analytic solutions when they exist.